The Mirror reports that Astori was on the shopping list before Dejan Lovren joined the Saints in January.
The 26-year-old has also been strongly linked with the likes of AC Milan and Everton.
The £10million-rated defender played for Italy in the Confederations Cup in Brazil last summer.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.